Method for controlling a wave power system by means of an integral proportional-control law

ABSTRACT

The present invention is a method of controlling a wave energy system (COM), wherein the force f ex  exerted by the waves on a mobile float of the wave energy system is estimated, then at least one dominant frequency ω ex  of the force exerted by the waves on the mobile float is determined using an unscented Kalman filter (UKF), and the control (COM) of the wave energy system is determined by a variable-gain PI control law whose coefficients are a function of dominant frequency ω ex .

CROSS REFERENCE TO RELATED APPLICATIONS

Reference is made to PCT/EP2018/054700 filed Feb. 26, 2018, designatingthe United States, and French Application No. 17/52.337 filed Mar. 22,2017, which are incorporated herein by reference in their entirety.

BACKGROUND OF THE INVENTION Field of the Invention

The invention relates to devices for converting wave energy intoelectrical or hydraulic energy.

Description of the Prior Art

Renewable energy resources have generated strong interest for years.They are clean, free and inexhaustible, which are major assets in aworld facing the inexorable depletion of the available fossil resourcesand recognizing the need to preserve the planet. Among these resources,the wave energy, a source relatively unknown amidst those widelypublicized, such as wind or solar energy, contributes to the vitaldiversification of the exploitation of renewable energy sources. Thedevices, commonly referred to as “wave energy devices”, are particularlyinteresting because they allow electricity to be produced from thisrenewable energy source (the potential and kinetic wave energy) withoutgreenhouse gas emissions. They are particularly well suited forproviding electricity to isolated island sites.

BACKGROUND OF THE INVENTION

For example, patent applications FR-2,876,751, FR-2,973,448 andWO-2009/081,042 describe devices intended to capture the energy producedby the sea water forces. These devices are made up of a floating supportstructure on which a pendulum movably mounted with respect to thefloating support is arranged. The relative motion of the pendulum withrespect to the floating support is used to produce electrical energywith an energy converter machine (an electrical machine for example).The converter machine operates as a generator and as a motor. Indeed, inorder to provide torque or a force driving a mobile floating device,power is supplied to the converter machine to bring it into resonancewith the waves (motor mode). On the other hand, to produce a torque or aforce that withstands the motion of the mobile floating device, power isrecovered via the converter machine (generator mode).

The motion of the mobile floating device is thus controlled by theenergy converter machine to promote energy recovery. In order tooptimize the electrical energy recovered by wave energy systems, variousconverter machine control methods have been considered. Some are notoptimal because the wave motion prediction is not taken intoconsideration. Furthermore, these methods do not take account for theenergy losses upon energy conversion in the wave energy system. Forexample, patent application FR-2,973,448 (WO-2012/131,186) describessuch a method.

PI control is a well-known approach for controlling wave energy systems.For example, the document by Ringwood, J. V., entitled ControlOptimisation and Parametric Design, in Numerical Modelling of WaveEnergy Converters: State-of-the-Art Techniques for Single WEC andConverter Arrays (M. Folly Ed.), Elsevier, 2016, describes such anapproach. Adaptive versions in relation to the sea state (that is whoseparameters vary depending on the sea state) have been presented in theliterature, for example in the following documents:

-   -   Whittaker, T., Collier, D., Folley, M., Osterried, M., Henry,        A., & Crowley, M. (2007, September). The Development of Oyster—a        Shallow Water Surging Wave Energy Converter. In Proceedings of        the 7th European Wave and Tidal Energy Conference, Porto,        Portugal (pp. 11-14).    -   Hansen, Rico H and Kramer, Morten M., “Modelling and Control of        the Wavestar Prototype”. In: Proceedings of 2011 European Wave        and Tidal Energy Conference (2011).    -   Hals, J., Falnes, J. and Moan, T., “A Comparison of Selected        Strategies for Adaptive Control of Wave Energy Converters”.        In: J. Offshore Mech. Arct. Eng. 133.3 (2011).

For all these methods, the approach used is gain scheduling, or gainpre-programming which is a set of optimal gains or parameters (one gainfor the P control, two gains for the PI control) which are calculatedoffline, analytically or numerically, for a set of sea states, tocomplete maps (charts) of gains as a function of the sea state. For themethods described in these documents, updating the PI gains is alwaysdone from averaged estimations, over time windows of several minutes(between 10 and 30 minutes for example). These methods therefore do notallow adapting the gains “wave by wave”, i.e. at a frequencycorresponding to the real-time control frequency (ranging between 10 and100 Hz for example, that is a sampling period ranging between 10 and 100ms). Thus, the high reaction time of these methods does not allowoptimal control of the wave energy system, and the recovered power istherefore not optimal. Furthermore, these methods do not allowoptimizing the power produced by the converter machine because they donot take the converter machine efficiency into account.

SUMMARY OF THE INVENTION

To overcome these drawbacks, the present invention provides a method ofcontrolling a wave energy system, wherein the force exerted by the waveson the mobile float of the wave energy system is estimated, then atleast one dominant frequency of the force exerted by the waves on themobile float is determined using an unscented Kalman filter (UKF), andthe control of the wave energy system is determined by a variable-gainPI control law whose coefficients are a function of the dominantfrequency. The control method according to the invention allows adaptingthe coefficients (gains) of the P and I actions, automatically andcontinuously, to the current sea state to maximize the power recoverablewith the PI control structure. This is done through the agency of anonline estimation, using an unscented Kalman filter (UKF), of thedominant frequency (or frequencies) of the wave spectrum.

The invention relates to a method of controlling a wave energy systemthat converts the energy of waves into electrical or hydraulic energy.The wave energy system comprises at least one mobile float in connectionwith an energy converter machine, and the mobile float has anoscillating motion with respect to the converter machine. For thismethod, the following steps are carried out:

-   -   a) measuring the position and/or the velocity and/or the        acceleration of the mobile float,    -   b) estimating the force exerted by the waves on the mobile float        using the measurement of at least one of the position and the        velocity of the mobile machine,    -   c) determining at least one dominant frequency of the force        exerted by the waves on said mobile float using an unscented        Kalman filter,    -   d) determining a control value of the force exerted by the        converter machine on the mobile float to maximize the power        generated by the converter machine, by use of a variable-gain        proportional integral PI control law whose coefficients are        determined by use of the dominant frequency of the force exerted        by the waves on the mobile float, and    -   e) controlling the converter machine by use of the control        value.

According to an embodiment, the proportionality coefficient Kp of the PIcontrol law is determined by use of an optimal load resistance curvefrom the dominant frequency of the force exerted by the waves on themobile float.

Advantageously, the integral coefficient Ki of the PI control law isdetermined by use of an optimal load reactance curve from the dominantfrequency of the force exerted by the waves on the mobile float.

Advantageously, the optimal load resistance Rc and the optimal loadreactance Xc are determined by solving the optimization problem asfollows:

${\min_{R_{c}X_{c}}\left\{ {{- \frac{R_{c}}{\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}}}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)} \right\}}\mspace{11mu}$  with $\mspace{20mu}\left\{ \begin{matrix}{{R_{i}(\omega)} = {B_{pa}(\omega)}} \\{{X_{i}(\omega)} = {\omega\left( {M + M_{\infty} + {M_{pa}(\omega)} - \frac{K_{pa}}{\omega^{2}}} \right)}}\end{matrix} \right.$where ω is the excitation frequency, B_(pa)(ω) is the radiationresistance of the mobile float, M the mass of the mobile float,M_(pa)(ω) is the added mass and M_(∞) is the infinite-frequency addedmass, K_(pa) is the hydrostatic stiffness of the mobile float, η_(p) isthe motor efficiency of the converter machine and η_(pn) is thegenerator efficiency of the converter machine.

Preferably, at least one of the optimal load resistance and the optimalload reactance curves are determined prior to carrying out the steps ofthe method.

According to an implementation, the power generated by the convertermachine is maximized by accounting for the efficiency of the convertermachine.

According to a characteristic, the PI control law is written with anequation of the type: f_(u)(t)=K_(p)v(t)+K_(i)p(t), with f_(u)(t) beingthe control of the force exerted by the converter machine on the mobilefloat, v(t) is the velocity of the mobile float, p(t) is the position ofthe mobile float, Kp is the proportionality coefficient of the PIcontrol law and Ki is the integral coefficient of the PI control law.

According to an embodiment, the control law is a variable-gain PIDproportional-integral-derivative control law.

Advantageously, the PID proportional-integral-derivative control law iswritten in the form as follows: f_(u)(t)=K_(p)v(t)+K_(i)p(t)+K_(d)a(t),with f_(u)(t) being the control of the force exerted by the convertermachine on the mobile float, v(t) being the velocity of the mobiledevice, p(t) being the position of the mobile float, a(t) being theacceleration of the mobile float, Kp being the proportionalitycoefficient of the PID control law, Ki being the integral coefficient ofthe PID control law and Kd being the derivative coefficient of the PIDcontrol law.

According to an embodiment option, the dominant frequency of the forceexerted by the waves on the mobile float is determined by modelling theforce exerted by the waves on the mobile float as a sine wave signal oras the sum of two sine wave signals.

According to an implementation of the invention, at least one of theposition and the velocity of the mobile float is estimated using adynamic model that represents the evolution of the position and of thevelocity of the mobile float.

Advantageously, the dynamic model comprises a model of the radiationforce.

Preferably, the energy converter machine is an electrical machine or ahydraulic machine.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the method according to the inventionwill be clear from reading the description hereafter, with reference tothe accompanying figures wherein:

FIG. 1 illustrates the steps of the method according to an embodiment ofthe invention;

FIG. 2 illustrates the determination of the control value according toan embodiment of the invention;

FIG. 3 illustrates a modelling of the wave energy system and of thecontrol thereof;

FIG. 4 illustrates a wave energy system according to an embodimentexample;

FIG. 5a illustrates an optimal load resistance curve for a firstapplication example;

FIG. 5b illustrates an optimal load reactance curve for the firstapplication example;

FIG. 6 illustrates a spectrum of one of the irregular waves for thefirst application example;

FIG. 7 illustrates an estimation of the dominant frequency of the waveforce by use of the control method according to the invention for thefirst application example;

FIG. 8 illustrates an estimation of the amplitude of the wave force byuse of the control method according to the invention for the firstapplication example;

FIG. 9 illustrates a comparative curve of the energy recovered with thecontrol method according to the invention and with a control methodaccording to the prior art for the first application example;

FIG. 10 illustrates a spectrum of an irregular wave for a secondapplication example; and

FIG. 11 illustrates a comparative curve of the energy recovered with thecontrol method according to the invention and with a control methodaccording to the prior art for the second application example.

DETAILED DESCRIPTION OF THE INVENTION

The invention relates to a method of controlling a wave energy systemthat comprises at least one mobile device such as a float for examplecooperating with at least one energy converter machine (also referred toas Power Take-Off PTO). The mobile float has an oscillating motion withrespect to the converter machine, under the action of the waves (or wavemotion) and of the converter machine. The converter machine converts themechanical energy of the motion of the mobile device into electricalenergy. The converter machine can therefore be a simple electricalmachine or a more complex device including other machines, such as ahydraulic machine. The converter machine can be considered as theactuator through which the control system drives the operation of thewave energy system.

Notations

The following notations are used in the description below:

-   -   f_(u): force exerted by the converter machine on the mobile        float,    -   f_(ex): force exerted by the waves on the mobile float,    -   p: position of the mobile float with respect to the equilibrium        point thereof,    -   v: velocity of the mobile float,    -   a: acceleration of the mobile float,    -   ω: frequency of the motion of the mobile float,    -   M: mass of the mobile float,    -   Z_(pa): radiation impedance, which is a parameter which is a        function of frequency and it is determined experimentally or        from the calculation of the hydrodynamic coefficients of the        mobile float, which allows accounting for the radiation        phenomenon, where the motion of the mobile float in the water        creates a radiated wave that dampens the float (that is which        dampens the motion),    -   K_(pa): hydrostatic stiffness coefficient,    -   B_(pa): radiation resistance, which is the real part of the        radiation impedance,    -   M_(pa): added mass, which parameter is a function of frequency        and is determined experimentally or from the calculation of the        hydrodynamic coefficients of the mobile float, which allows        accounting for the phenomenon that increases the equivalent mass        of the mobile float, due to the water particles carried along by        the motion thereof,    -   M_(∞): infinite-frequency added mass,    -   P_(a): average power generated by the wave energy system,    -   t: continuous time,    -   S: Laplace variable,    -   k: discrete time,    -   η: energy conversion efficiency, with        -   η_(p): motor efficiency of the converter machine; these are            manufacturer's data or experimentally determined data,        -   η_(n): generator efficiency of the converter machine; these            are manufacturer's data or experimentally determined data,    -   Z_(i): intrinsic impedance of the mobile float of the wave        energy system, this known parameter is a function of frequency        and it results from modelling the mobile device on the basis of        the linear wave theory, determined from the hydrodynamic        coefficients of the mobile means and possibly experimental        measurements,    -   R_(i): intrinsic resistance of the mobile float of the wave        energy system, i.e. the real part of the intrinsic impedance,    -   X_(i): intrinsic reactance of the mobile float of the wave        energy system, i.e. the imaginary part of the intrinsic        impedance,    -   K_(p): proportionality coefficient of the PI control law,    -   K_(i): integral coefficient of the PI control law,    -   K_(d): derivative coefficient of the PID control law,    -   Z_(c): control law (or load) impedance,    -   R_(c): control law (or load) resistance, with        -   R_(v) ^(D): optimal resistance for a given sea state,    -   X_(c): control law reactance, with        -   X_(c) ⁰: optimal resistance for a given sea state,    -   A: wave motion amplitude,    -   {circumflex over (ω)}_(ex): estimation of the dominant frequency        of the wave,    -   {circumflex over (f)}_(ex): estimation of the force exerted by        the wave on the mobile float,    -   T_(s): sampling period,    -   ϕ: signal phase shift,    -   A_(x): state model matrix,    -   C: state model matrix,    -   v(k): Gaussian noise with covariance matrix Q,    -   μ(k): Gaussian noise with covariance matrix R,    -   P_(x): covariance matrix of x(k).

For these notations, the estimated values are generally written with ahat. Time is denoted by t (continuous variable) or k (discretevariable).

In the description below and for the claims, the terms waves, sea wavesand wave motion are considered to be equivalent.

The invention relates to a method of controlling a wave energy system.FIG. 1 shows the various steps of the method according to the invention:

1. Measurement of the position and/or the velocity of the mobile float(p, v)

2. Estimation of the force exerted by the waves (EST)

3. Determination of the dominant frequency (UKF)

4. Determination of the control value (COEFF)

5. Control of the converter machine (COM).

Steps 1 to 5 are carried out in real time, in a real-time loop. However,according to an embodiment of the invention, determination of thecontrol value can comprise calculating optimal load resistance andreactance curves beforehand.

Advantageously, the control method according to the invention can beimplemented by performing computation by, a computer for example, or aprocessor, in particular an on-board processor.

Step 1—Measurement of the Position and/or the Velocity of the MobileFloat (p, v)

The position and/or the velocity of the mobile float are measured inthis step. The position corresponds to the motion (distance or angle forexample) with respect to the equilibrium position of the mobile float.These measurements can be performed using sensors, generally implementedon a wave energy system for control and/or supervision thereof.

According to an implementation of the invention, in this step, it isalso possible to measure or to estimate the acceleration of the mobilefloat, and this measurement or estimation can be used in the next stepsof the method according to the invention. For example, the accelerationcan be measured using an accelerometer arranged on the mobile float.

Step 2—Estimation of the Force Exerted by the Waves (EST)

In this step, the force exerted by the waves on the mobile float isestimated in real time. Estimation of the wave force is performed fromthe available measurements (position and/or velocity and/oracceleration) obtained in the previous step. The force exerted by thewaves on the mobile float is estimated online and in real time to enablereal-time control. A fast estimation method can be selected with a viewto control with an optimal response time.

For this step of the method, any type of estimation of the force exertedby the waves on the mobile float may be considered.

According to an embodiment of the invention, the force exerted by thewaves on the mobile float can be estimated using an estimator based on adynamic model of the wave energy system. In this case, a dynamic modelof the wave energy system can be constructed. The dynamic modelrepresents the dynamic behaviour due to the motion of the elements thatmake up the wave energy system under the action of the waves and underthe action of the force command given to the converter machine. Thedynamic model is a model relating the velocity of the mobile float tothe force exerted by the waves on the mobile float and to the forcecommand given to the converter machine, which is in turn translated intoa force exerted by the converter machine on the mobile float.

According to an embodiment of the invention, the dynamic model can beobtained by applying the fundamental principle of dynamics to the mobilefloat of the wave energy system. For this application, the force exertedby the waves on the mobile float and the force exerted by the convertermachine on the mobile float are notably taken into account.

According to an implementation of the invention, a wave energy systemwith a floating part (mobile float) whose translational or rotationaloscillating motion is constrained in a single dimension may beconsidered. It is then assumed that the translational or rotationalmotion can be described by a linear model in form of a state includingthe dynamics of the float with its interaction with the waves and thedynamics of the power take-off (PTO) system, or converter machine,forming the actuator of the system.

In the rest of the description below, only a unidirectional motion isconsidered for the dynamic model. However, the dynamic model can bedeveloped for a multidirectional motion.

According to an example embodiment, the force exerted by the waves onthe mobile float can be estimated in real time using a method ofdetermining the excitation force exerted by the incident waves on amobile float of a wave energy system, by use of a model of the radiationforce, as described in patent application No. FR-16/53,109. As areminder, the radiation force is the force applied onto the mobile floatand is generated by the very motion of the mobile float, unlike the waveexcitation force that is generated by the waves only.

Step 3—Determination of the Dominant Frequency

In this step, at least one dominant frequency of the force exerted bythe waves on the mobile float (determined in the previous step) isdetermined. What is referred to as dominant frequency is the frequencycorresponding to the peak (maximum) of the spectrum thereof. Thedominant frequency is determined using an unscented Kalman filter (UKF).An unscented Kalman filter is based on the unscented transformationtheory which provides an estimator for a non-linear system withouthaving to linearize it beforehand applying the filter. The UKF filteruses a statistical state distribution that is propagated through thenon-linear equations. Such a filter affords the advantage of providingstability, and therefore robustness of the estimation.

According to an embodiment of the invention, the excitation force of thewave is modelled as a time-varying sine wave signal:f _(ex)(t)=A(t)sin(ω(t)t+ϕ(t))where A(t), ω(t) and ϕ(t) are the amplitude, the frequency and the phaseshift of the signal respectively. It is an approximation because thisforce is not in reality a varying-parameter sinusoid. In the linear wavetheory, it is rather modelled as a superposition of constant-parametersinusoids.

Alternatively, the excitation force of the wave can be modelled as thesum of two time-varying sine wave signals:f _(ex)(t)=A ₁(t)sin(ω₁(t)+ϕ₁(t))+A ₂(t)sin(ω₂(t)+ϕ₂(t))

These (time-varying) wave excitation force modelling parameters need tobe estimated. Since they are entered non-linearly into the aboveequation, it is a non-linear estimation problem.

The unscented Kalman filter method is used to estimate A(t), ω(t) andϕ(t).

It is noted that other non-linear estimation methods could in principlebe used to carry out this step, such as the extended Kalman filter (EKF)or the particle filter, but the UKF gives particularly good results. Theuse of the EKF filter in particular has been mentioned in theliterature; however, as it is based on the local linearization of anon-linear model, it does not guarantee the same estimation stabilityand therefore robustness as the UKF filter.

To apply the UKF filter, the equation modelling the excitation force isfirst put in discrete-time state form.

Let T_(s) be the sampling period of the filter. The wave excitationforce is then estimated in discrete time t=kT_(s), k=0, 1, 2, . . . ,which is simply denoted by k.

By defining

$\left\{ {\begin{matrix}{{x_{1}(k)} = {A\;{\sin\left( {{{kT}_{s}\omega} + \phi} \right)}}} \\{{x_{2}(k)} = {A\;{\cos\left( {{{kT}_{s}\omega} + \phi} \right)}}} \\{{x_{3}(k)} = \omega}\end{matrix},} \right.$and assuming that ω(k) changes slowly over time (in relation to thesampling period), the model can be obtained in state form as follows:

$\left\{ {\begin{matrix}{{\begin{bmatrix}{x_{1}(k)} \\{x_{2}(k)} \\{x_{3}(k)}\end{bmatrix} = {{\begin{bmatrix}{\cos\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & {\sin\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & 0 \\{- {\sin\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)}} & {\cos\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}{x_{1}\left( {k - 1} \right)} \\{x_{2}\left( {k - 1} \right)} \\{x_{3}\left( {k - 1} \right)}\end{bmatrix}} + \begin{bmatrix}{v_{1}\left( {k - 1} \right)} \\{v_{2}\left( {k - 1} \right)} \\{v_{3}\left( {k - 1} \right)}\end{bmatrix}}},} \\{{f_{ex}(k)} = {{\begin{bmatrix}1 & 0 & 0\end{bmatrix}\begin{bmatrix}{x_{1}(k)} \\{x_{2}(k)} \\{x_{3}(k)}\end{bmatrix}} + {\mu(k)}}}\end{matrix}\quad} \right.$

In this model, uncertainties were added to take account for modellingerrors. More particularly, v₁(k−1) and v₂(k−1) serve to make up for thetime-varying nature of A(t), ω(t) and ϕ(t) that is not taken intoaccount in the definition of the states x₁(k) and x₂(k). v₃(k−1) is anuncertainty of the third state (the frequency to be estimated), which isnaturally correlated with v₁(k−1) and v₂(k−1). μ(k) is an uncertaintythat can be equated with a measurement error on f_(ex)(k), which servesto account for the fact that f_(ex)(k) is not exactly a sinusoid.

By noting:

${{x(k)} = \begin{bmatrix}{x_{1}(k)} & {x_{2}(k)} & {x_{3}(k)}\end{bmatrix}^{T}},{{v(k)} = \begin{bmatrix}{v_{1}(k)} & {v_{2}(k)} & {v_{3}(k)}\end{bmatrix}^{T}},{{y(k)} = {f_{ex}(k)}}$ and $\left\{ {\begin{matrix}{A_{x} = \begin{bmatrix}{\cos\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & {\sin\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & 0 \\{- {\sin\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)}} & {\cos\left( {T_{s}{x_{3}\left( {k - 1} \right)}} \right)} & 0 \\0 & 0 & 1\end{bmatrix}} \\{C = \begin{bmatrix}1 & 0 & 0\end{bmatrix}}\end{matrix},} \right.$the equations of the state model are written as follows:

$\left\{ {\begin{matrix}{{x(k)} = {{A_{x}{x\left( {k - 1} \right)}} + {v(k)}}} \\{{y(k)} = {{{Cx}(k)} + {\mu(k)}}}\end{matrix},} \right.$The following assumptions are adopted:

-   -   the initial state x(0) is a random vector of mean m(0)=E[x(0)]        and of covariance P(0)=E[(x(0)−m(0))(x(0)−m(0))^(T)],    -   v(k) and μ(k) are Gaussian noises with covariance matrices Q and        R respectively,        as well as the following notations:    -   x(k|k−1) is the estimation of x(k) from measurements up to the        time k−1, that is y(k−1), y(k−2), . . . .    -   x(k|k) is the estimation of x(k) from measurements up to the        time k, that is y(k), y(k−1) . . . .    -   P_(x)(k|k−1) is the covariance matrix of x(k) from measurements        up to the time k−1, that i y(k−1), y(k−2), . . . .    -   P_(x)(k|k) is the covariance matrix of x(k) from measurements up        to the time k, that is y(k), y(k−1) . . . .        There are three steps in the UKF method:

1. Calculation of the Sigma Points

In this first step, a set of samples in the state space is calculatedwhich are referred to as sigma points, and indeed represent theprobabilistic distribution of the state according to the mean andcovariance parameters thereof.

Let:

${W_{0}^{m} - \frac{\lambda}{n + \lambda}},{W_{0}^{c} = {\frac{\lambda}{n + \lambda} + \left( {1 - \alpha^{2} + \beta} \right)}},{W_{j}^{m} = {W_{j}^{c} = \frac{\lambda}{2\left( {n + \lambda} \right)}}},{j = 1},2,\ldots\mspace{14mu},{2n}$where λ=(α²−1)n is a scaling parameter, α is a parameter that determinesthe spread of the sigma points around x(k−1|k−1), which is generallyassigned a small positive value, 10⁻³ for example, β is a parameter usedto incorporate a priori knowledge on the distribution of x: for aGaussian distribution, β=2 which is optimal.

At the time k−1, the following selection is considered of sigma points(set of points encoding exactly the mean and covariance information):x ₀(k−1)=x(k−1|k−1),x _(i)(k−1)=x(k−1|k−1)+√{square root over (n+λ)}S _(i)(k−1),i=1,2, . . .,nx _(i+n)(k−1)=x(k−1|k−1)−√{square root over (n+λ)}S _(i)(k−1),i=1,2, . .. ,nwhere S_(i)(k−1) is the i-th column of the matrix square root ofP_(x)(k−1|k−1), that is P_(x)(k|k−1)=S(k−1)_(T)S(k−1).

2. Prediction Updating

Each sigma point is propagated through the non-linear model representingthe evolution of the states:x _(j)(k|k−1)=A _(x) x _(j)(k−1),j=0,1, . . . ,2n

The mean and the covariance of {circumflex over (x)}(k|k−1), theprediction of x(k|k−1){circumflex over (x)}(k|k−1)=Σ_(j=0) ^(2n) W _(j) ^(m) {circumflex over(x)} _(j)(k|k−1),P _(x)(k|k−1)=Σ_(j=0) ^(2n) W _(j) ^(c)({circumflex over (x)}_(j)(k|k−1)−x(k|k−1))({circumflex over (x)} _(j)(k|k−1)−x(k−k−1))^(T) +Q

The predicted states {circumflex over (x)}_(j)(k|k−1) are used in theoutput state equation, which yields:ŷ _(j)(k|k−1)=C{circumflex over (x)} _(j)(k|k−1)

The mean and the covariance of ŷ(k|k−1) are calculated as follows:ŷ(k|k−1)=Σ_(j=0) ^(2n) W _(j) ^(m) ŷ _(j)(k|k−1),P _(y)(k|k−1)−Σ_(j=0) ^(2n) W _(j) ^(c)(ŷ _(j)(k|k−1)−y(k|k−1))(ŷ_(j)(k|k−1)−y(k|k−1))^(T) +Rwhile the cross-covariance between {circumflex over (x)}(k|k−1) andŷ(k|k−1) is:

${P_{xy}\left( k \middle| {k - 1} \right)} = {\sum\limits_{j = 0}^{2n}{{W_{j}^{c}\left( {{{\hat{x}}_{j}\left( k \middle| {k - 1} \right)} - {x\left( k \middle| {k - 1} \right)}} \right)}\left( {{{\hat{y}}_{j}\left( k \middle| {k - 1} \right)} - {y\left( k \middle| {k - 1} \right)}} \right)^{T}}}$

3. Updating from the Measurements

As in the Kalman filter, the final state estimation is obtained bycorrecting the prediction with a feedback on the output prediction error(measured):{circumflex over (x)}(k)={circumflex over (x)}(k|k−1)+K(ŷ(k)−ŷ(k|k−1))where gain K is given by:K=P _(xy)(k|k−1)P _(y)(k|k−1)⁻¹

The a posteriori covariance estimation is updated with the formula asfollows:P _(x)(k|k)=P _(x)(k|k1)KP _(y)(k|k1)K ^(T)

Step 4—Determination of the Control Value (VAL)

In this step, the control value of the force exerted by the convertermachine on the mobile float is determined to maximize the powergenerated by the converter machine. This determination is performedusing a variable-gain proportional-integral control law whosecoefficients (the variable gains) are determined as a function of thedominant frequency of the force exerted by the waves on the mobilefloat.

According to an implementation of the invention, the control law can bea variable-gain proportional-integral-derivative type control law.

According to an embodiment, a force-velocity dynamic model can bewritten for this type of wave energy system in frequency form:

${\left( {{j\;\omega\; M} + {Z_{pa}\left( {j\;\omega} \right)} + \frac{K_{pa}}{j\;\omega}} \right){v\left( {j\;\omega} \right)}} = {{f_{ex}\left( {j\;\omega} \right)} - {f_{u}\left( {j\;\omega} \right)}}$where:

-   -   v(jω) is the velocity of the mobile float,    -   f_(ex)(jω) and f_(u)(jω) are the excitation force of the        incident wave and the force applied by the converter machine        onto the mobile float, respectively,    -   M is the mass of the mobile float (float for example) and of all        the other parts of the wave energy system secured to this mobile        float,    -   Z_(pa)(jω) is the radiation impedance,    -   K_(pa) is the hydrostatic stiffness.

This model is obtained from the Cummins integro-differential equation,and its coefficients K_(pa) and Z_(pa)(jω) (and those resulting from itsdecomposition, below) can be calculated using hydrodynamic codes basedon the boundary element method (BEM), such as WAMIT, Diodore or NEMOH.The radiation impedance Z_(pa)(jω) which, in the linear wave theory,describes the effect of the free motion of the float in the water, isthe result of an approximation of the radiation impulse response of aninfinite impulse response filter. It can be decomposed as follows:

$\begin{matrix}{{Z_{pa}\left( {j\;\omega} \right)} = {{B_{pa}\left( {j\;\omega} \right)} + {j\;{\omega\left( {{M_{pa}\left( {j\;\omega} \right)} + M_{\infty}} \right)}}}} \\{= {{H_{pa}\left( {j\;\omega} \right)} + {j\;\omega\; M_{\infty}}}}\end{matrix}$where B_(pa)(jω) is the radiation resistance, M_(pa)(jω) is the addedmass after removal of the infinite singularity M_(∞) andH_(pa)(jω)=B_(pa)(jω)+jωM_(pa)(jω).

The velocity of the float, as a function of the forces applied thereon,can be rewritten as follows:

${v\left( {j\;\omega} \right)} + {\frac{1}{Z_{j}\left( {j\;\omega} \right)}\left( {{f_{ex}\left( {j\;\omega} \right)} - {f_{u}\left( {j\;\omega} \right)}} \right)}$where the intrinsic impedance Z_(i)(jω) is defined as:

${Z_{i}\left( {j\;\omega} \right)} = {{B_{pa}(\omega)} + {j\;{\omega\left( {M + M_{\infty} + {M_{pa}(\omega)} - \frac{K_{pa}}{\omega^{2}}} \right)}} - {R_{t}(\omega)} + {{jX}_{t}(\omega)}}$  where $\mspace{20mu}\left\{ \begin{matrix}{{R_{i}(\omega)} = {B_{pa}(\omega)}} \\{{X_{i}(\omega)} = {\omega\left( {M + M_{\infty} + {M_{pa}(\omega)} - \frac{K_{pa}}{\omega^{2}}} \right)}}\end{matrix} \right.$are respectively the intrinsic resistance and reactance (real part andimaginary part of the impedance) of the system.

In this step, optimization of the performances of the control law interms of maximization of the electrical power produced on average P_(a):

$P_{a} = {\frac{1}{T}{\int_{t = 0}^{T}{\eta\;{f_{u}(t)}{v(t)}{dt}}}}$where η is a coefficient representing the converter machine efficiency.If η=1, the converter machine is considered to be perfect, with noenergy conversion losses. Although this assumption is unrealistic, it isoften adopted in the literature because it greatly simplifies thecalculations, in this case for the optimal parameters of a PI typecontrol law. It however amounts to considering that drawing power fromthe grid through the converter machine costs the same as deliveringpower, which is generally wrong and may lead to a much lower electricalenergy production than expected, or even to grid energy waste (P_(a)negative).

According to an implementation, efficiency r_(i) can be considered to bea function of the instantaneous power f_(u) ^(v) defined as follows:

${\eta\left( {f_{u}v} \right)} = \left\{ \begin{matrix}{{\eta_{p}\mspace{14mu}{if}\mspace{14mu} f_{u}v} \geq 0} \\{{\eta_{n}\mspace{14mu}{if}\mspace{14mu} f_{u}v} < 0}\end{matrix} \right.$where coefficients 0<η_(p)≤1 and η_(n)≥1 depend on the converter machineand may even be a function of f_(u) ^(v).

This step is a PI control law for hydrodynamic control of a wave energysystem:f _(u)(t)=K _(p) v(t)+K _(i) p(t)=K _(p) v(t)+K _(i)∫₀ ^(r) v(τ)dτthat is, in the Laplace domain:

${{f_{u}(s)} - {K_{p}{v(s)}} + {\frac{K_{i}}{s}{v(s)}}},{s - \sigma + {j\;\omega}}$whose parameters K_(p) and K_(i) are continuously adapted, that isonline and in real time, as a function of the estimated dominantfrequency of the excitation force of the wave, so as to guarantee thatthe electrical power produced is maximized for this frequency.

Alternatively, the control law is a PID law that can be written for thecontrol of a wave energy system:f _(u)(t)=K _(p) v(t)+K _(i) p(t)+K _(d) a(t)

Maximization of the electrical power produced is based on an analytical(original) expression relating, frequency by frequency. The power of thereal part (resistance) and of the imaginary part (reactance) of theimpedance achieved by the control law, assuming that the force appliedby the converter machine is a linear feedback on the velocity of themobile float of the wave energy system (see FIG. 3) may be expressed as:f _(u)(s)=Z _(c)(s)v(s)

After denoting by

$\left\{ {\begin{matrix}{R_{c} = {{Re}\left\{ Z_{c} \right\}}} \\{X_{c} = {{Im}\left\{ Z_{c} \right\}}}\end{matrix}\quad} \right.$the resistance and the reactance of the control law (or load) impedanceat a given frequency (the dominant frequency of the wave excitationforce), the analytical expression for the electrical power at thisfrequency is:

$P_{a} = {\frac{A^{2}R_{c}}{2\left( {\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}} \right)}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)}$where P_(i) and X_(i) are calculated at the frequency in question fromthe model of the wave energy system. Parameter A can be obtained fromthe wave force estimation (but, as shown hereafter, it is not necessaryto calculate it), and R_(c) and X_(c) need to be determined for the samefrequency by solving the optimization problem as follows:

$\min_{R_{c}X_{c}}\left\{ {{- \frac{R_{c}}{\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}}}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)} \right\}$which amounts to maximizing the power defined above. It is noted thatparameter A can be left out since it does not influence the optimalsolution.

Since this optimization problem is non-linear, there is no closedanalytical solution thereto but it can be solved numerically. Offline,parameters R_(c) and X_(c) can therefore be calculated numerically foreach frequency in the interval of interest, to obtain curves connectedto the dominant frequency of the wave force. These curves are queriedonline in order to obtain the optimal parameters R_(c) ^(o),X_(c) ^(o)from {circumflex over (ω)}_(ex), the estimation of the dominantfrequency of the wave.

The optimal gains of the PI control law are then calculated as:

$\left\{ {\begin{matrix}{K_{p}^{o} = R_{c}^{o}} \\{K_{i}^{o} = {{- {\hat{\omega}}_{ex}}X_{c}^{o}}}\end{matrix}\quad} \right.$

The control step is schematized by way of non-limitative example in FIG.2. At the input of this step of determining the control value, there areestimations of the dominant frequency of the wave excitation force{circumflex over (ω)}_(ex), and the measurements of the velocity v(t)and the position p(t) of the mobile float. By use of a curve C1, anoptimal load resistance curve that can be obtained offline, and thedominant frequency of the wave excitation force {circumflex over(ω)}_(ex), the optimal load resistance R_(c) ^(o) corresponding to theproportional coefficient of the PI control law K_(p) ^(o) is determined.In parallel, by use of a curve C2, an optimal load reactance curve canbe obtained offline, and of the dominant frequency of the waveexcitation force {circumflex over (ω)}_(ex), the optimal load reactanceX_(c) ^(o) is determined. The integral coefficient of the PI control lawK_(i) ^(o) is obtained by multiplying the optimal load reactance X_(c)^(o) by the dominant frequency of the wave excitation force {circumflexover (ω)}_(ex). The control law f_(u)(t) is then obtained by adding themultiplication of the proportional coefficient K_(p) ^(o) by thevelocity v(t) to the multiplication of integral coefficient K_(i) ^(o)by the position p(t).

Alternatively, it is possible to first perform offline the constructionof curves of optimal coefficients K_(i) and K_(p) as a function of thedominant frequency of the wave force, and then to determine onlinecoefficients K_(i) and K_(p) from these curves and the dominantfrequency exerted by the wave force. Thus, the control law is completelydefined.

According to the embodiment where the control law is a PID control law,the variable coefficient Kd of the control law can be obtained using theestimation of a second dominant frequency (by modelling the waves withtwo sinusoids for example).

Step 5—Control of the Converter Machine

In this step, the converter machine is controlled as a function of thevalue determined in the previous step. The converter machine (electricalor hydraulic machine) is therefore actuated to reproduce the new valueof force f_(u) as determined in step 4.

For example, the new expression for the control of force u allowingobtaining a force f_(u) exerted by the converter machine on the mobilefloat is applied to the control system of the electrical machine.Controlling the electrical machine so that it applies the correspondingforce f_(u), down to the dynamics of the machine, to the requestedcontrol u is achieved by modifying, if need be, the electrical currentapplied to the electrical machine. More precisely, to provide a torqueor a force that drives the mobile float, a current is applied bysupplying electrical power. On the other hand, to produce a torque or aforce withstanding the motion of the mobile float, a current is appliedby recovering electrical power.

A non limitative example of a wave energy system is an oscillating buoyas shown in FIG. 4. This wave energy system comprises a buoy 2 as themobile float of mass m, a converter machine 1 with its control law,whose action f_(PTO) can be represented by a damping d and an elasticityk. The buoy is subjected to an oscillating motion through waves 3 andthe force f_(PTO) applied by the converter machine. Converter machine 1can be an electrical machine connected to an electric grid 4.

APPLICATION EXAMPLE

The features and advantages of the method according to the inventionwill be clear from reading the application example hereafter.

In this example, a float as described in FIG. 4 whose force-velocitydynamics (velocity response to the sum of the forces applied on thefloat) is given by the transfer function as follows:

${Z_{i}(s)} = \frac{\begin{matrix}{s^{6} + {208.6s^{5}} + {{8.583 \cdot 10^{4}}s^{4}} + {{8.899 \cdot 10^{6}}s^{3}} + {1.074 \cdot}} \\{{10^{8}s^{2}} + {{7.031 \cdot 10^{8}}s}}\end{matrix}}{\begin{matrix}{{1.44s^{7}} + {300.4s^{6}} + {{1.237 \cdot 10^{5}}s^{5}} + {{1.284 \cdot 10^{7}}s^{4}} +} \\{{{1.652 \cdot 10^{8}}s^{3}} + {{2.106 \cdot 10^{9}}s^{2}} + {{9.988 \cdot 10^{9}}s} + {6.539 \cdot 10^{10}}}\end{matrix}}$

The transfer function describes the dynamics of a small-scale prototype(1:20) to which the method was applied.

The converter machine efficiency parameters taken into account are:η_(p)=η_(n)=0.7. With these parameters and the above transfer functionZ_(i)(s), the following optimization problem can be solved:

${\min\limits_{R_{c}X_{c}}\left\{ {{- \frac{R_{c}}{\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}}}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)} \right\}},{R_{c} \geq 0}$and the two optimal load resistance and reactance curves of FIGS. 5a and5b used for online calculation of the parameters of the PI control laware obtained.

FIG. 5a illustrates, as a function of frequency ω, the optimalresistance R_(o) resulting from the previous optimization. This curveshows in dotted line the intrinsic resistance Ri and in solid line theoptimal resistance R_(o).

FIG. 5b illustrates, as a function of frequency ω, the optimal reactanceX_(o) resulting from the previous optimization. This curve shows indotted line the intrinsic reactance Xi and in solid line the optimalreactance X_(o).

In order to validate the method, the wave energy system driven by theadaptive PI control law according to the invention was subjected to aseries of irregular wave tests. The spectrum S (Nm²s/rad) of one ofthese waves, slightly longer than 1000 s and with a dominant frequencyof approximately 5 rad/s, is shown in FIG. 6.

The good performance of the online step of estimating the dominantfrequency of the force of wave w (rad/s) as a function of time t(s) withthe control method according to the invention is shown in FIG. 7.

FIG. 8 illustrates the amplitude A of wave wa and the amplitude Aest(dark line) estimated with the control method according to theinvention. This figure shows that the UKF algorithm used in the controlmethod according to the invention to estimate the parameters of thevariable-parameter sinusoidal model of the wave excitation force canmanage great transitions (towards t=1040 s in the figure for example)while allowing to obtain stable estimations for amplitude A(t) anddominant frequency ω(t), thanks to the robustness thereof overnon-linearities. Other estimation approaches, in particular EKF, wouldnot be capable of such a performance.

FIG. 9 illustrates the energy recovered by the adaptive PI controlaccording to the invention INV, and by a non-adaptive PI controlaccording to the prior art AA. It is the energy recovered by the PIcontrol according to the invention or according to the prior art withfixed parameters on an irregular wave. Thus, this curve compares theenergy recovery performances, for the same wave, of the adaptive PIcontrol law according to the invention and of a PI control law withfixed parameters optimized from the characteristics of the sea statespectrum, according to a state of the art. It is noted that the PIcontrol law according to the invention even allows recovery of moreenergy than in a situation where fixed parameters according to the priorart should be sufficient.

A second test was carried out with an irregular wave whose dominantfrequency varies linearly between the frequency of the first wave tested(5 rad/s) and that of a wave of frequency 1.8 rad/s. FIG. 10 shows thespectrum S (Nm²s/rad) of the irregular wave as a function of frequency w(rad).

FIG. 11 is a curve similar to the curve of FIG. 9 for the irregular wavewhose spectrum is illustrated in FIG. 10. As shown in FIG. 11, themethod according to the invention INV allows recovery of much moreenergy than a PI control with fixed parameters calibrated on the firstspectrum, according to a prior art.

Therefore, the control method according to the invention provides anonline control that optimizes the recovered energy.

The invention claimed is:
 1. A method of controlling a wave energysystem that converts energy of waves into electrical or hydraulicenergy, the wave energy system comprising at least one mobile float inconnection with an energy converter machine, and the mobile float havingan oscillating motion with respect to the converter machine, comprisingsteps of: a) measuring at least one of position, velocity andacceleration of the mobile float, b) estimating force exerted by thewaves on the mobile float using the measurement of at least one of theposition and velocity of the mobile float, c) determining at least onedominant frequency of the force exerted by the waves on the mobile floatusing an unscented Kalman filter; d) determining a control value of theforce exerted by the converter machine on the mobile float to maximizepower generated by the converter machine, by use of a variable-gainproportional integral PI control law to provide the control law withcoefficients determined by use of at least one of the at least onedominant frequency of the force exerted by the waves on the mobilefloat; and e) controlling the converter machine by use of the controlvalue.
 2. A method as claimed in claim 1, wherein a proportionalitycoefficient of the PI control law is determined by use of an optimalload resistance curve from at least one of the at least one dominantfrequency of the force exerted by the waves on the mobile float.
 3. Amethod as claimed in claim 1, wherein an integral coefficient of the PIcontrol law is determined by use of an optimal load reactance curve fromat least one of the at least one dominant frequency of the force exertedby the waves on the mobile float.
 4. A method as claimed in claim 2,wherein an integral coefficient of the PI control law is determined byuse of an optimal load reactance curve from at least one of the at leastone dominant frequency of the force exerted by the waves on the mobilefloat.
 5. A method as claim 2, wherein the optimal load resistance Rcand the optimal load reactance Xc are determined by solving anoptimization problem as follows:$\min_{R_{c}X_{c}}{\left\{ {{- \frac{R_{c}}{\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}}}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)} \right\}\mspace{11mu}{with}}$$\mspace{20mu}\left\{ \begin{matrix}{{R_{i}(\omega)} = {B_{pa}(\omega)}} \\{{X_{i}(\omega)} = {\omega\left( {M + M_{\infty} + {M_{pa}(\omega)} - \frac{K_{pa}}{\omega^{2}}} \right)}}\end{matrix} \right.$ where ω is the excitation frequency, B_(pa)(ω) isradiation resistance of the mobile float, M is mass of the mobile float,M_(pa)(ω) is added mass and M_(∞) is infinite-frequency added mass,K_(pa) is hydrostatic stiffness of the mobile float, η_(p) is motorefficiency of the converter machine and η_(pn) is generator efficiencyof the converter machine.
 6. A method as claim 3, wherein the optimalload resistance Rc and the optimal load reactance Xc are determined bydetermining an optimization as follows:$\min_{R_{c}X_{c}}{\left\{ {{- \frac{R_{c}}{\left( {X_{c} + X_{i}} \right)^{2} + \left( {R_{c} + R_{i}} \right)^{2}}}\left( {\eta_{p} - {\frac{\eta_{n} - \eta_{p}}{\pi}\left( {\frac{X_{c}}{R_{c}} - {\arctan\left( \frac{X_{c}}{R_{c}} \right)}} \right)}} \right)} \right\}\mspace{11mu}{with}}$$\mspace{20mu}\left\{ \begin{matrix}{{R_{i}(\omega)} = {B_{pa}(\omega)}} \\{{X_{i}(\omega)} = {\omega\left( {M + M_{\infty} + {M_{pa}(\omega)} - \frac{K_{pa}}{\omega^{2}}} \right)}}\end{matrix} \right.$ where ω is the excitation frequency, B_(pa)(ω) isradiation resistance of the mobile float, M is mass of the mobile float,M_(pa)(ω) is added mass and M_(∞) is infinite-frequency added mass,K_(pa) is hydrostatic stiffness of the mobile float, η_(p) is motorefficiency of the converter machine an η_(pn) is generator efficiency ofthe converter machine.
 7. A method as claimed in claim 2, wherein atleast one of the optimal load resistance and optimal load reactancecurves are determined prior to performing the method.
 8. A method asclaimed in claim 5, wherein at least one of the optimal load resistanceand optimal load reactance curves are determined prior to performing themethod.
 9. A method as claimed in claim 1, wherein the power generatedby the converter machine is maximized accounting for efficiency of theconverter machine.
 10. A method as claimed in claim 1, wherein the PIcontrol law is expressed as an equation: f_(u)(t)=K_(p)v(t)+K_(i)p(t),with f_(u)(t) being the control of the force exerted by the convertermachine on the mobile float, v(t) is the velocity of the mobile float,p(t) is position of the mobile float, Kp is a proportionalitycoefficient of the PI control law and Ki is an integral coefficient ofthe PI control law.
 11. A method as claimed in claim 1, wherein thecontrol law is a variable-gain PID proportional-integral-derivativecontrol law.
 12. A method as claimed in claim 11, wherein the PIDproportional-integral-derivative control law is expressed as follows:f_(u)(t)=K_(p)v(t)+K_(i)p(t)+K_(d)a(t), with f_(u)(t) being the controlof the force exerted by the converter machine on the mobile float, v(t)being velocity of the mobile float, p(t) being the position of themobile float, a(t) being the acceleration of the mobile float, Kp beingthe proportionality coefficient of the PID control law, Ki being theintegral coefficient of the PID control law and Kd being the derivativecoefficient of the PID control law.
 13. A method as claimed in claim 1,wherein the dominant frequency of the force exerted by the waves on themobile float is determined by modelling the force exerted by the waveson the mobile float as a sine wave signal or a sum of two sine wavesignals.
 14. A method as claimed in claim 1, wherein the position andthe velocity of the mobile float is estimated using a dynamic modelrepresenting evolution of the position and of velocity of the mobiledevice.
 15. A method as claimed in claim 14, wherein the dynamic modelcomprises a model of radiation force.
 16. A method as claimed in claim1, wherein the energy converter machine is an electrical machine or ahydraulic machine.
 17. A method as claimed in claim 2, wherein thecontrol law is a variable-gain PID proportional-integral-derivativecontrol law.
 18. A method as claimed in claim 3, wherein the control lawis a variable-gain PID proportional-integral-derivative control law. 19.A method as claimed in claim 4, wherein the control law is avariable-gain PID proportional-integral-derivative control law.
 20. Amethod as claimed in claim 5, wherein the control law is a variable-gainPID proportional-integral-derivative control law.